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ArcSech




Mathematica Notation

Traditional Notation









Elementary Functions > ArcSech[z] > Representations through equivalent functions > With related functions > Involving cot-1 > Involving sech-1((2(1+z2)1/2/((1+z2)1/2-z))1/2) > Involving sech-1((2(1+z2)1/2/((1+z2)1/2-z))1/2) and cot-1(z)





http://functions.wolfram.com/01.30.27.1072.01









  


  








Input Form





ArcSech[Sqrt[(2 Sqrt[1 + z^2])/(Sqrt[1 + z^2] - z)]] == (Pi/4) Sqrt[1/z] Sqrt[-z] (1 + Sqrt[1/z^2] z Sqrt[1/(1 + z^2)] Sqrt[1 + z^2]) - (1/2) Sqrt[1/z] Sqrt[-z] ArcCot[z]








Standard Form





Cell[BoxData[RowBox[List[RowBox[List["ArcSech", "[", SqrtBox[FractionBox[RowBox[List["2", SqrtBox[RowBox[List["1", "+", SuperscriptBox["z", "2"]]]]]], RowBox[List[SqrtBox[RowBox[List["1", "+", SuperscriptBox["z", "2"]]]], "-", "z"]]]], "]"]], "\[Equal]", RowBox[List[RowBox[List[FractionBox["\[Pi]", "4"], " ", SqrtBox[FractionBox["1", "z"]], SqrtBox[RowBox[List["-", "z"]]], RowBox[List["(", RowBox[List["1", "+", RowBox[List[SqrtBox[FractionBox["1", SuperscriptBox["z", "2"]]], " ", "z", " ", SqrtBox[FractionBox["1", RowBox[List["1", "+", SuperscriptBox["z", "2"]]]]], " ", SqrtBox[RowBox[List["1", "+", SuperscriptBox["z", "2"]]]]]]]], ")"]]]], "-", RowBox[List[FractionBox["1", "2"], " ", SqrtBox[FractionBox["1", "z"]], SqrtBox[RowBox[List["-", "z"]]], RowBox[List["ArcCot", "[", "z", "]"]]]]]]]]]]








MathML Form







<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <msup> <mi> sech </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <msqrt> <mfrac> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msqrt> <mrow> <mn> 1 </mn> <mo> + </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> </msqrt> </mrow> <mrow> <msqrt> <mrow> <mn> 1 </mn> <mo> + </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> </msqrt> <mo> - </mo> <mi> z </mi> </mrow> </mfrac> </msqrt> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <mrow> <mfrac> <mi> &#960; </mi> <mn> 4 </mn> </mfrac> <mo> &#8290; </mo> <msqrt> <mfrac> <mn> 1 </mn> <mi> z </mi> </mfrac> </msqrt> <mo> &#8290; </mo> <msqrt> <mrow> <mo> - </mo> <mi> z </mi> </mrow> </msqrt> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <msqrt> <mfrac> <mn> 1 </mn> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mfrac> </msqrt> <mo> &#8290; </mo> <msqrt> <mfrac> <mn> 1 </mn> <mrow> <msup> <mi> z </mi> <mn> 2 </mn> </msup> <mo> + </mo> <mn> 1 </mn> </mrow> </mfrac> </msqrt> <mo> &#8290; </mo> <msqrt> <mrow> <msup> <mi> z </mi> <mn> 2 </mn> </msup> <mo> + </mo> <mn> 1 </mn> </mrow> </msqrt> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mfrac> <msqrt> <mrow> <mo> - </mo> <mi> z </mi> </mrow> </msqrt> <mn> 2 </mn> </mfrac> <mo> &#8290; </mo> <msqrt> <mfrac> <mn> 1 </mn> <mi> z </mi> </mfrac> </msqrt> <mo> &#8290; </mo> <mtext> </mtext> <mrow> <msup> <mi> cot </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <arcsech /> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <plus /> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <times /> <pi /> <apply> <power /> <cn type='integer'> 4 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <plus /> <apply> <times /> <apply> <power /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <plus /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <plus /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> 1 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <ci> z </ci> </apply> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <times /> <apply> <power /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <arccot /> <ci> z </ci> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["ArcSech", "[", SqrtBox[FractionBox[RowBox[List["2", " ", SqrtBox[RowBox[List["1", "+", SuperscriptBox["z_", "2"]]]]]], RowBox[List[SqrtBox[RowBox[List["1", "+", SuperscriptBox["z_", "2"]]]], "-", "z_"]]]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[FractionBox["1", "4"], " ", "\[Pi]", " ", SqrtBox[FractionBox["1", "z"]], " ", SqrtBox[RowBox[List["-", "z"]]], " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List[SqrtBox[FractionBox["1", SuperscriptBox["z", "2"]]], " ", "z", " ", SqrtBox[FractionBox["1", RowBox[List["1", "+", SuperscriptBox["z", "2"]]]]], " ", SqrtBox[RowBox[List["1", "+", SuperscriptBox["z", "2"]]]]]]]], ")"]]]], "-", RowBox[List[FractionBox["1", "2"], " ", SqrtBox[FractionBox["1", "z"]], " ", SqrtBox[RowBox[List["-", "z"]]], " ", RowBox[List["ArcCot", "[", "z", "]"]]]]]]]]]]


Date Added to functions.wolfram.com (modification date)





2003-08-21